{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-finite-markov-chain"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-finite-markov-chain-01","title":{"zh-Hant":"有限狀態 Markov 鏈：核對行隨機矩陣，計算兩步時刻律與路徑機率","en":"Finite-state Markov chain: checking a row-stochastic matrix and computing two-step time laws and path probabilities"},"summary":{"zh-Hant":"狀態依序為 1、2、3，轉移矩陣 P 的行為 (1/2,1/2,0)、(1/4,1/2,1/4)、(0,1/3,2/3)，初始律 μ_0=(1/2,1/2,0)。(1) 核對 P 是行隨機矩陣，並說明列和不必為 1；(2) 證明一步公式 μ_{n+1}=μ_nP 且結果仍是機率向量，算出 μ_1 與 μ_2；(3) 證明路徑機率公式並計算 P(X_0=1,X_1=2,X_2=3)、P(X_0=2,X_1=3,X_2=3)；(4) 說明 P 單獨一般不能決定完整的無條件時刻律與路徑律。","en":"The states are ordered 1, 2, 3; the transition matrix P has rows (1/2,1/2,0), (1/4,1/2,1/4), (0,1/3,2/3), and the initial law is μ_0=(1/2,1/2,0). (1) Check that P is row-stochastic and explain why column sums need not be 1; (2) prove the one-step formula μ_{n+1}=μ_nP and that the result is again a probability vector, and compute μ_1 and μ_2; (3) prove the path-probability formula and compute P(X_0=1,X_1=2,X_2=3) and P(X_0=2,X_1=3,X_2=3); (4) explain why P alone generally does not determine complete unconditional marginal and path laws."},"mko_ids":["mko-finite-markov-chain"]}],"title":"Concept derivation guides"}